I got the parts directly from John Deere, so I sure hope they are correct! Being that the new carb is acting exactly like the old one though, I wouldn't think the new and old have the same issues by fluke. But I'll try to check the part # tomorrow. I checked the spark with inline tester again today. I think the tool itself isn't the best. Cheapo from princess auto. It seems to "miss" the odd time on both sides - but not how I've seen misses before in automotive where it clearly doesn't spark, and you can hear the miss at the same time. This would backfire and pop randomly with no sign of missing spark, then "potentially" lose spark (I don't trust tool) with no sign of backfire or running rough. Basically the visual cues not lining up at all with the audio cues of backfire. I did get the fan and shroud off, and verified that the key is not sheared off - DARN. Would have been an easy fix. My bro is currently installing RC14YC plugs, as we read somewhere that the 12s may not run hot enough. We figured what's the harm? Worst case is he has a spare set of plugs for in a few years.
I looked it up on the JD website. It has their part numbers, rather than briggs. But a search brought up this website that lists the JD part number and says it is the OEM replacement for 791230 like you say. So I suspect it is right.
https://www.greenpartstore.com/John-Deere-Carburetor-Assembly-MIA10632.html
I'm kind of at a stalemate. I don't want to throw more money at it, even though it's not mine. So if I take the heads off - costs new head gaskets at minimum. And unless there is a glaring problem, I may be stuck going that route and not finding anything. Or, I can suspect the coils are getting weak and replace them, probably for $$$. Which to me, "shouldn't" cause backfire, only misfire. - though I never rule anything out.
This thing is eating my lunch. Ironically I keep finding things wrong, but the issue never gets solved. (crazy loose terminal at solenoid, though wow, that should fix hard start after clean and tighten. NOPE)(Backfiring, and hard start? 90% chance worn camshaft and failed decompression valve - NOPE not equipped, and wear too minor)(also needs an ignition switch because sometimes you turn it and nothing at all until you wiggle or tap it, with clean connections)
I will be calling my friend at JD to see if he knows anything to look for. I will also test my spark tester on a known good engine to see how accurate tool is, since it's relatively new to me still. Otherwise I'm crossing my fingers on these spark plugs. Otherwise I may spend another few hours staring blankly at it trying to convince myself its anything but the valves/heads